Download Lecture Notes for Section 2.1

Trigonometry Lecture Notes
Section 2.1
Page 1 of 5
Section 2.1: Trigonometric Functions of Acute Angles
Big Idea: If we restrict an angle in standard position to be an acute angle, then a right triangle is
formed by dropping a vertical from the point on the terminal side to the x axis
Big Skill: You should be able to .
Right Triangle Based Definitions of Trigonometric Functions (Section 2.1)
(SOH CAH TOA)
r  x2  y 2
y opp b


r hyp c
y opp b
tan   

x adj a
r hyp c
sec   

x adj a
sin  
c  a 2  b2
x adj a
cos   

r hyp c
x adj a
cot   

y opp b
r hyp c
csc   

y opp b
Cofunction Identities (Section 2.1)
For any acute angle A,
sin A  cos  90  A
sec A  csc  90  A
tan A  cot  90  A
cos A  sin  90  A
csc A  sec  90  A
cot A  tan  90  A
Trigonometry Lecture Notes
Section 2.1
Page 2 of 5
Practice:
1. Practice the right triangle definitions of an acute angle using a 36 – 77 – 85 right triangle,
and a right triangle with sides of length 1 and 2.
2. Show why the cofunction identities are true.
3. Solve for : cot   8  tan  4  13
4. State how the sine, cosine, and tangent functions increase or decrease with increasing
angle.
Trigonometry Lecture Notes
Section 2.1
Page 3 of 5
Trigonometric Function Values of Special Angles
The trigonometric Values of a 45 angle can be derived by thinking of an isosceles right triangle.
Compute the length of the hypotenuse for the triangle below, and then use the side lengths to
derive all trigonometric function values for an angle of 45
Trigonometric Function Values for 45 (Section 2.1)
opp

hyp
opp
tan 45 

adj
hyp
sec 45 

adj
sin 45 
adj

hyp
adj
cot 45 

opp
hyp
csc 45 

opp
cos 45 
Trigonometry Lecture Notes
Section 2.1
Page 4 of 5
The trigonometric values of 30 and 60 angles can be derived by thinking of the triangles
formed when an equilateral triangle is bisected. Compute the length of the altitude formed by
bisecting the equilateral triangle below, and then use the side lengths to derive all trigonometric
function values for angles of 30 and 60.
Trigonometric Function Values for 30 and 60 (Section 2.1)
opp

hyp
opp
tan 60 

adj
hyp
sec 60 

adj
opp
sin 30 

hyp
opp
tan 30 

adj
hyp
sec30 

adj
sin 60 
adj

hyp
adj
cot 60 

opp
hyp
csc 60 

opp
adj
cos 30 

hyp
adj
cot 30 

opp
hyp
csc30 

opp
cos 60 
Trigonometry Lecture Notes
Section 2.1
Page 5 of 5
The trigonometric values of 36 and 72 angles can be derived by starting with a 36-72-72
isosceles triangle, bisecting one of the base angles, deriving relationships between the side
lengths using similar triangles, and then bisecting the apex angle to form right triangles. Perform
these calculations using the 36-72-72 isosceles triangle below, and then use the side lengths to
derive all trigonometric function values for angles of 36 and 72.
Trigonometric Function Values for 36 and 72 (Section 2.1 Mirus Special)
opp

hyp
opp
tan 72 

adj
hyp
sec 72 

adj
opp
sin 36 

hyp
opp
tan 36 

adj
hyp
sec36 

adj
sin 72 
adj

hyp
adj
cot 72 

opp
hyp
csc 72 

opp
adj
cos 36 

hyp
adj
cot 36 

opp
hyp
csc36 

opp
cos 72 